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An Asymptotic (4/3+ε)-Approximation for the 2-Dimensional Vector Bin Packing Problem
[article]

2022
*
arXiv
*
pre-print

We give

arXiv:2205.12828v1
fatcat:bo4rqxmmk5fvpk3xz5kyk4mu7a
*an**asymptotic*(*4*/*3*+*ε*)-*approximation**for**the**problem*, thus improving upon*the*best known*asymptotic*ratio of (1+ln3/*2*+ε)≈ 1.406 due to Bansal, Elias and Khan (SODA 2016). ... We study*the**2*-*Dimensional**Vector**Bin**Packing**Problem*(2VBP), a generalization of classic*Bin**Packing*that is widely applicable in resource allocation and scheduling. ... Our main result is*an*improved*asymptotic*approximation ratio*for**the**problem*. Theorem 1.1.*For*any ε > 0 there is a polynomial-time random*asymptotic**4**3*+*ε*-*approximation**for*2VBP. ...##
###
Tight Approximation Algorithms for Two Dimensional Guillotine Strip Packing
[article]

2022
*
arXiv
*
pre-print

Moreover, due to a reduction from

arXiv:2202.05989v3
fatcat:bzrmvh3gmbbjlpw6jdd5xn3p4y
*the*Partition*problem*, it is NP-hard to obtain a polynomial-time (3/*2*-ε)-approximation algorithm*for*GSP*for*any ε>0 (exactly as Strip*Packing*). ... This*problem*generalizes*the*classical*Bin**Packing**problem*and also makespan minimization on identical machines, and thus it is already strongly NP-hard. ... There is a pseudo-polynomial time (*4*/*3*+*ε*)-*approximation*[22]*for*2GK. ...##
###
Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items
[article]

2021
*
arXiv
*
pre-print

*The*conjecture, if true, will imply a (

*4*/

*3*+

*ε*)-

*approximation*algorithm

*for*2BP. ... In

*the*Two-

*dimensional*

*Bin*

*Packing*(2BP)

*problem*, we are given a set of rectangles of height and width at most one and our goal is to find

*an*axis-aligned nonoverlapping

*packing*of these rectangles into ...

*An*

*Asymptotic*Polynomial-Time Approximation Scheme (APTAS) is

*an*algorithm that accepts a parameter ε and has AAR of (1 + ε).

*2*BP is a generalization of classical 1-D

*bin*

*packing*

*problem*[22, 15] . ...

##
###
Approximation Algorithms for Rectangle Packing Problems (PhD Thesis)
[article]

2017
*
arXiv
*
pre-print

Thus, we consider a generalized kind of

arXiv:1711.07851v1
fatcat:awwgt5x74fbfzim5mmsmndrlhu
*packing*that combines container*packings*with another*packing**problem*that we call L-*packing**problem*, where we have to*pack*rectangles in*an*L-shaped region of*the*... In*2*-*Dimensional*Geometric Knapsack (2DGK),*the*target region is a square of a given size, and our goal is to select and*pack*a subset of*the*given rectangles of maximum weight. ... Chapter 3 A PPT (*4*/*3*+*ε*)-*approximation**for*Strip*Packing*In this chapter, we present a (*4**3*+*ε*)-*approximation**for*Strip*Packing*running in pseudo-polynomial time. ...##
###
Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More
[article]

2021
*
arXiv
*
pre-print

In this paper, we give (

arXiv:2103.10406v1
fatcat:gn6xqsyugjc37ic53xret5xxlm
*4*/*3*+*ε*)-*approximation*algorithms in polynomial time*for*both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.' ... In*the**2*-*Dimensional*Knapsack*problem*(2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits. ... , so that there exists a feasible*packing*of a (*4*/*3*+*ε*)-*approximate*solution into them. ...##
###
A PTAS for Packing Hypercubes into a Knapsack
[article]

2022
*
arXiv
*
pre-print

*For*this

*problem*, Harren (ICALP'06) gave

*an*algorithm with

*an*approximation ratio of (1+1/

*2*^d+epsilon). ... As a side result, we give

*an*almost optimal algorithm

*for*a variant of

*the*strip

*packing*

*problem*in higher dimensions. ... Acknowledgements We thank Roberto Solis-Oba and three anonymous reviewers

*for*their comments and suggestions on this paper. ...

##
###
Approximating Geometric Knapsack via L-packings
[article]

2017
*
arXiv
*
pre-print

We study

arXiv:1711.07710v1
fatcat:wyl6uu4dercwxldztvrphpfdki
*the*two-*dimensional*geometric knapsack*problem*(2DK) in which we are given a set of n axis-aligned rectangular items, each one with*an*associated profit, and*an*axis-aligned square knapsack. ...*The*best-known polynomial-time approximation factor*for*this*problem*(even just in*the*cardinality case) is (*2*+ \epsilon) [Jansen and Zhang, SODA 2004]. ... Open*Problems**The*main*problem*that we left open is to find a PTAS, if any,*for*2DK and 2DKR. This would be interesting even in*the*cardinality case. ...##
###
Closing the gap for pseudo-polynomial strip packing
[article]

2019
*
arXiv
*
pre-print

*The*set of

*2*-

*dimensional*

*packing*

*problems*builds

*an*important class of optimization

*problems*and Strip

*Packing*together with

*2*-

*dimensional*

*Bin*

*Packing*and

*2*-

*dimensional*Knapsack is one of

*the*most famous ...

*The*strength of this structural result is that it applies to other

*problem*settings as well

*for*example to Strip

*Packing*with rotations (90 degrees) and Contiguous Moldable Task Scheduling. ... On

*the*other hand,

*the*algorithm with

*the*best ratio so far computes a

*4*/

*3*+

*ε*

*approximation*[8, 16] . ...

##
###
Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

2021

In this paper, we give (

doi:10.4230/lipics.socg.2021.39
fatcat:u2za5n3xfzbnjf7z7zczzoe7iy
*4*/*3*+*ε*)-*approximation*algorithms in polynomial time*for*both cases, assuming that all input data are integers polynomially bounded in n. Gálvez et al.' ... In*the**2*-*Dimensional*Knapsack*problem*(2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits. ... so that there exists a feasible*packing*of a (*4*/*3*+*ε*)-*approximate*solution into them. ...##
###
Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

2021

*The*conjecture, if true, will imply a (

*4*/

*3*+

*ε*)-

*approximation*algorithm

*for*2BP. ... In

*the*Two-

*dimensional*

*Bin*

*Packing*(2BP)

*problem*, we are given a set of rectangles of height and width at most one and our goal is to find

*an*axis-aligned nonoverlapping

*packing*of these rectangles into ...

*Vector*

*packing*(VP) is another variant of

*bin*

*packing*, where

*the*input is a set of

*vectors*in [0, 1] d and

*the*goal is to partition

*the*

*vectors*into

*the*minimum number of parts (

*bins*) such that in each ...

##
###
A PTAS for Packing Hypercubes into a Knapsack

2022

*For*this

*problem*, Harren (ICALP'06) gave

*an*algorithm with

*an*approximation ratio of (1+1/

*2*^d+ε). ... As a side result, we give

*an*almost optimal algorithm

*for*a variant of

*the*strip

*packing*

*problem*in higher dimensions. ... Acknowledgements We thank Roberto Solis-Oba and three anonymous reviewers

*for*their comments and suggestions on this paper. ...